terms; and the interest on 1 month's wages, $.10, is the common difference; and since the first month's wages has been on interest 17 months, the progression is a descending series. Then, by 706 we find the first term, which is the amount of the first month's wages for 17 months; and by 709 we find the sum of the series, which is the sum of all the wages and interest. 2. A father deposits annually for the benefit of his son, commencing with his tenth birthday, such a sum that on his 21st birthday the first deposit at simple interest amounts to $210, and the sum due his son to $1860. How much is the deposit, and at what rate per cent. is it deposited? terms; and $1860, the amount of all the deposits and interests, is the sum of the series. By 709 we find the last term to be $100, which is the annual deposit; and by 707 we find the common difference to be $10, which is the annual rate %. 3. What is the amount of an annuity of $150 for 5 years, payable quarterly, at 1 per cent. per quarter? Ans. $3819.75. 4. In what time will an annual pension of $500 amount to $3450, at 6 per cent. simple interest? Ans. 6 years. 5. Find the rate per cent at which an annuity of $6000 will amount to $59760 in 8 years, at simple interest. Ans. 7 per cent. ANNUITIES AT COMPOUND INTEREST. 727. An Annuity at compound interest constitutes a geometrical progression whose first term is the annuity itself; the common multiplier is one plus the rate per cent. for one interval expressed decimally; the number of terms is the number of intervals for which the annuity is taken; and the last term is the first term multiplied by one plus the rate per cent. for one interval raised to a power one less than the number of terms. 728. The Present Value of an Annuity is such a sum as would produce, at compound interest, at a given rate, the same amount as the sum of all the payments of the annuity at compound interest. Hence, to find the present value;-First find the amount of the annuity at the given rate and for the given time by 711; then find the present value of this amount by 714, or 715. NOTES.-1. The present value of a reversionary annuity is that principal which will amount, at the time the reversion expires, to what will then be the present value of the annuity. 2. The present value of a perpetuity is a sum whose interest equals the an- ~ nuity. 729. Questions in Annuities at compound interest can be solved by the rules of Geometrical Progression. PROMISCUOUS EXAMPLES IN SERIES. 1. Allowing 6 per cent. compound interest on an annuity of $200 which is in arrears 20 years, what is its present amount? Ans. $7357.11. 2. Find the annuity whose amount for 25 years is $16459.35, allowing compound interest at 6 per cent. Ans. $300. 3. What is the present worth of an annuity of $500 for 7 years, Ans. $2791.18. at 6 per cent. compound interest? 4. What is the present value of a reversionary lease of $100, commencing 14 years hence, and to continue 20 years, compound interest at 5 per cent.? Ans. $629.426. 5. Find the sum of 21 terms of the series, 5, 44, 44, etc. 526. A man traveled 13 days; his last day's journey was 80 miles, and each day he traveled 5 miles more than on the preceding day. How far did he travel, and what was his first day's journey? Ans. He traveled 650 miles. 7. Find the 12th term of the series, 30, 15, 71, etc. Ans. 15 8. The first term of a geometrical progression is 2, the last term. 512, and common multiplier 4; find the sum of the series. Ans. 682. 5=405 408 +2 = 8167 8 = 102 102 - 42 = 60 ÷ 2 = 63044 SERIES. 2 416 30 9. The distance between two places is 360 miles. In how many days can it be traveled, by a man who travels the first day 27 miles, and the last day 45, each day's journey being greater than the preceding by the same number of miles? Ans. 10. 10. The first term of a geometrical progression is 1, the last term 15625, and the number of terms 7; find the common ratio. Ans. 5. What 11. An annual pension of $500 is in arrears 10 years. is the amount now due, allowing 6 per cent. compound interest? Ans. $6590.40. 12. Find the first and last terms of an arithmetical progression whose sum is 408, common difference 6, and number of terms 8. Ans. First term, 30; last term, 72. 13. A farmer pays $1196, in 13 quarterly payments, in such a way that each payment is greater than the preceding by $12. What are his first and last payments? Ans. $20, and $164. 14. A man wishes to discharge a debt in such a way that his first payment shall be $2, his last $512, and each payment four times the preceding payment. How long will it take him to discharge the debt, and what is the amount of his indebtedness? 15. A man dying, left 5 sons, to whom he gave his property as follows to the youngest he gave $4800, and to each of the others 11⁄2 times the next younger son's share. What was the eldest son's fortune, and what the amount of property left? Ans. Eldest son's share, $24300; property, $63300. 16. Find the annuity whose amount for 5 years, at 6 per cent. compound interest, is $2818.546. 02 Ans. $2232.552+. 17. A merchant pays a debt in yearly payments in such a way that each payment is 3 times the preceding; his first payment is $10, and his last $7290. What is the amount of the debt, and in how many payments is it discharged? Ans. Debt, $10930; 7 payments. 18. A man traveling along a road, stopped at a number of stations, but at each station he found it necessary, before proceeding to the next, to return to the place from whence he first started; the distance from the starting place to the first station was 5 miles, and to the last 25 miles; he traveled in all 180 miles. many stations were there on the road, and what was the distance from station to station? Ans. 5 stations; 4 miles apart. 19. An annuity of $200 for 12 years is in reversion 6 years. What is its present worth, compound interest at 6 %? Ans. 1182.05+. 20. A man pays $6 yearly for tobacco, from the age of 16 until he is 60, when he dies, leaving to his heirs $500. What might he have left them, if he had dispensed with this useless habit and loaned the money at the end of each year at 6 % compound interest? Ans. 1698.548+. 21. What is the present worth of a reversionary perpetuity of $100, commencing 30 years hence, allowing 5 per cent. compound interest? 462,75 22. Two boys, each 12 years old, have certain sums of money left to them; the sum left to one is put out at 7 % simple interest, payable annually, and the sum left the other at 6 % compound interest, payable semi-annually, and the amount of each boy's money will be $2000 when he is 21 years old. What is the sum left to each boy? 1 7 4 7 8 7 23. A merchant purchased 8 pieces of cloth, for which he paid $136; the difference in the length of any two pieces was 2 yds., and the difference in the price $4. He paid $31 for the longest piece, and $1 a yard for the shortest. Find the whole number of yards, and the price per yard of each piece. 24. A farmer has 600 bushels of different kinds of grain, mixed in such a way that the number of bushels of the several kinds constitute a geometrical progression, whose common multiplier is 2; the greatest number of bushels of one kind is 320. Find the number of kinds of grain in the mixture, and the number of bushels of each kind. Ans. 4 kinds. 2B MISCELLANEOUS EXAMPLES. 1. How many thousand shingles will cover both sides of a roof 36 ft. long, and whose rafters are 18 ft. in length? /3 2. From of of of 70 miles, subtract .73 of 1 mi. 3 fur. 3. What number is that from which if 71⁄2 be subtracted, remainder is 91? of the Ans. 1443. Ans. . 4. What part of 4 is of 6? 5 It is required to mix together brandy at $.80 a gallon, wine at $.70, cider at $.10, and water, in such proportions that the mixture may be worth $.50 a gallon; what quantity of each must be used? Ans. 9 gal, each of brandy and wine; 5 gal. each of cider and water. 6. What number increased by 1, 3, and 4 of itself equals 125? 7. What is the hour, when the time past noon is equal to of the time to midnight? Ans. 4 h. 48 min. P. M. 8. A grocer mixed 12 cwt. of sugar @ $10, with 3 cwt. @ $83, and 8 cwt. @$73; how much was 1 cwt. of the mixture worth? f 3 9. If $240 gain $5.76 in 4 mo. 26 da,, what is the rate %? Ans. 6. 10. If 24 men, in 189 da., working Ph. a day, dig a trench 334 yd. लु long, 3 yd. deep, and 53 yd. wide; how many hours a day must 217 men work, to dig a trench 234 yd. long, 24 yd. deep, and 3 yd. wide, in 5 days? Ans. 16 h. 11. What is the difference between the interest and the discount of $450 at 5 per cent., for 6 yr. 10 mo.? 39.1534 12. A younger brother received $6300, which was as much as his elder brother received; how much did both receive? 200. 13. Reduce .7, .88, 727, .91325 to their equivalent common fractions. 14. A person by selling a lot of goods for $438, loses 10 %; how much should the goods have been sold for, to gain 123 %? 547,50 15. For what sum must a note be drawn at 4 mo., that the proceeds of it, when discounted at bank at 7 per cent., shall be $875.50? 16. Three persons engaged in trade with a joint capital of $2128; A's capital was in trade 5 m, B's 8 mo., and C's 12 mo.; A's share of the gain was $228, B's $255.40, and C's $330. What was the capital of each? Ins. A's, $912; B's, $666; C's, $550. 17. Henry Truman purchased corn of John Bates, on 2 months' credit, as follows: Aug. 27, 300 bu. @ $.35; Aug. 31, 150 bu. @ $.40; Sept. 7, 500 bu. @@ $.38; Sept. 12, 200 bu. (@ $,42; Sept. 25, 250 bu, @$.40. When was the a/c, due per average?V. Ans. Sept. 8. 18. A B and C can do a job of work in 12 da., C can do it in 24 da., and A in 34 da.; in what time can B do it alone? da. 19. If a man travel 7 mi. the first day, and 51 mi. the last, increasing his journey 4 mi. each day, how many days will he travel, and how far? Ans. 12 da., and 348 mi. Ans. 81 |